A car-following framework for traffic instability and lane changes

TL;DR

This work addresses traffic instability and lane-changing behavior by integrating Newell's first-order car-following with a density-dependent stability criterion and a psychology-based lane-changing rule. It shows a critical reaction time $τ$ that depends on vehicle density, with the continuum limit recovering Newell's result, and demonstrates load-balancing across lanes via a stochastic lane-changing mechanism that ties driver frustration to change attempts. The findings reveal that aggressive driving increases lane-change frequency with modest distance benefits, highlighting safety trade-offs, and offer a framework linking microscopic driver psychology to macroscopic traffic stability. The approach provides quantitative insights relevant to traffic management and autonomous-vehicle design, suggesting avenues for further development in second-order dynamics and empirical calibration.

Abstract

This paper develops a computational framework based on a car-following model to study traffic instability and lane changes. Building upon Newell's classical first-order car-following model, we show that, both analytically and numerically, there exists a vehicle-density-dependent critical reaction time that determines the stability of single-lane traffic. Specifically, perturbations to the equilibrium system decay with time for low reaction time and grow for high reaction time. This critical reaction time converges to Newell's original result in the continuum limit. Additionally, we propose a psychology-based lane-changing mechanism that builds a quantitative connection between the driver's psychological factor (frustration level) and the driving condition. We show that our stochastic lane-changing model can faithfully reproduce interesting phenomena like load-balancing of different lanes. Our model supports the result that more frequent lane changes only marginally benefit the driver's overall velocity.

Figures (7)

• Figure 1: (a) The driving velocity $\dot{x}_j$ as a function of headway $h_j \equiv x_{j+1} - x_j$ given by \ref{['eq:velocity-eq']}. $V$ is the maximal velocity, $d$ is the minimal headway for a non-zero velocity, and $\lambda$ is the rate of change of the velocity at $h_j=d$. In the equilibrium state where all vehicles have the same headway $h_\infty$, the equilibrium velocity of each vehicle $v_\infty$ can be found directly via \ref{['eq:velocity-eq']}. (b) The equilibrium flow rate $q$ as a function of vehicle density $\rho$.
• Figure 2: Schematic figure of our car-following model. Each rectangle denotes a vehicle whose position $x_{j, l}$ is represented by the dot at the center of the rectangle. The width of each rectangle represents the size of the vehicle $C$, and the length of the gray whisker denotes the minimal headway $d$. Red rectangles denote the vehicles whose drivers are accumulating frustration because the headway in the adjacent lane is bigger than the headway in its own lane. Filled rectangles denote the vehicles that have enough gap in the adjacent lane to complete a lane change (no car in [$x_{j,l} - d, x_{j,l} + d$]), while open rectangles do not. The simulation is done in a primary cell of length $L$. All positions are measured from the left end of the primary cell. Vehicles in the image cells are shown in lighter colors, as a reminder that they are merely images of the vehicles of the primary cell.
• Figure 3: Flowchart of the numerical method. The steps are grouped into Lane-changing, Collision-detecting, and Forward-moving stages, denoted by blue, red, and green colors respectively.
• Figure 4: Dynamics of 50 vehicles in a single lane with various reaction times $\Delta = 0, 0.5, 0.75$s. (a-c) Velocity as a function of time for representative vehicles with different reaction times. Curves of blue, orange, purple, red, and green represent the results of vehicles 1, 11, 21, 31, and 41 respectively. Dashed curves denote the decay/growth of velocity difference. (d-f) The flow rate at different position $x$ and time $t$ for various reaction time $\Delta$. The time step and the averaging time window for all simulations are $\Delta t = 0.01s, \delta = 18.63s$.
• Figure 5: Stability of a single-lane traffic system. (a) The cyclic growth rate of velocity difference $k$ of the non-linear simulations and the maximal real part of the characteristic roots of the linearized system as functions of reaction time $\Delta$. (b) The critical reaction time $\tau$ as a function of the number of vehicle in the system $N$.